On a problem posed by Y.-Z. Huang

Doplicher-Roberts theorem describes gauge symmetries commuting with the observables as compact groups in the setting of algebraic QFT. In conformal field theory Moore and Seiberg conjectured that rational theories give rise to modular tensor categories in the sense of Reshetikhin and Turaev. They also conjectured to understand chiral algebras as generalizations of quantum groups. The construction of modular tensor categories has a complicated history. In the algebraic approach to CFT the notion of conformal net appeared in the work by Buchholz, Mack and Todorov. Many people gave important contributions, and modularity was established by Kawahigashi, Longo, and Mueger. A culmination of the work by many people including Drinfeld, Jimbo, Andersen, Kirillov, Wenzl provides unitary modular fusion categories from quantum groups at roots of unity. The combination of the work by Kazhdan, Lusztig, Finkelberg, gives the construction of a braided tensor category for affine Lie algebras at positive integer level, thereby establishing a tensor equivalence with the quantum group setting, but the proof is indirect. A direct construction of modular tensor category in the setting of affine vertex operator algebras was obtained by Huang and Lepowsky. Huang has recently posed the problem of establishing a direct proof of KLF equivalence theorem. Mack and Schomerus proposed weak quasi-Hopf algebras as quantum symmetries extending Drinfeld quasi-Hopf algebras. In my talk I would like to discuss an approach to FKL Theorem via a class of weak Hopf algebras extending Woronowicz compact quantum groups.